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Representing complex engineering systems by simple mathematical models May 23, 2007 Jason Mayes Outline Motivation Mathematical modeling Complex systems Objectives Present Work Using self-similarity Using fractional-order system identification Future Work Mathematical Modeling A mathematical model is a mathematical description of a physical system or process Design Prediction Control Mathematical modeling: finding a compromise between an intractable problem and a model that sufficiently describes the system Characteristics of a desirable model Simple Adequately describes the system Fd xo Mathematical Modeling x(t) • Electro-mechanical model • Equations of motion • Add complexity Fg • Experiments Random Experimental Data 1.2 1 0.8 0.6 t 0.4 0.2 0 0 1 2 3 4 5 6 Xo The philosophy of analysis Many ways to find a mathematical model Methodological Reductionism Descartes‟ 1637 Discourse on Method The world is like a machine… Everything can be reduced to many smaller, simpler things The best way to understand a system is to first gain a clear understanding of its smallest subsystems Models are based on first principles Problems: Size and complexity Holism Behavior must be studied on the level of the system as a whole Aristotle‟s Metaphysics – “the whole is more than the sum of its parts” Random Experimental Data Examples 1.2 1 Neural nets 0.8 On/off, PID, fuzzy logic 0.6 t 0.4 Expert systems 0.2 0 0 1 2 3 4 5 6 Xo Black-box analysis is useful for describing or controlling systems, but it doesn‟t explain observed behavior A model-based compromise Mix of holistic and reductionist approaches Model-based parameter identification Can form a model containing a few free parameters Very common in heat transfer N Nusselt number correlations, convection F N Contact resistance, fouling coefficients Heat exchangers Friction factor (Moody diagram) F Trend in science: HolisticReductionist analysis Example: chemical reactions Complex systems Q: What is a complex system? 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Properties: Size/complexity Non-linear Emergent phenomena Memory Our working definition: complex engineering systems Physically: any system composed of a large number of components and interactions that creates difficulties in both understanding and modeling Mathematically: a large system of coupled equations which are either too complex or too large to admit a sufficiently useful solution Objectives To find simple mathematical models of complex engineering systems Using both reductionist and holistic approaches to modeling Systems with similarity Both physical and mathematical similarity Can reduce complex mathematical models to simpler models Using fractional-order system identification Modeling complex systems as a black-box Better results than traditional integer-order models Mathematical self-similarity Current Work: using a reductionist approach and taking advantage of mathematical similarity to simplify complex models Mathematical similarity: Equations of the same form Repeating patterns or coupling in equations Reducible mathematical models Complex system: large system of ODEs System of first order ODEs scalar ODE High-order scalar ODE lower order scalar ODE Mathematical Similarity Reducing infinite-order ODEs High-order ODEs can be reduced in the Laplace domain Geometric Series!! Mathematical Similarity Reducing infinite-order ODEs Can reduce an infinite-order (very large) ODE to a simple, finite-order ODE Only useful in complex mechanical systems if infinite-order ODEs occur in modeling Mathematical Similarity Reducing infinite sets of ODEs High-order (infinite) ODEs result from large (infinite) equations sets Mathematical Similarity Reducing infinite sets of ODEs Consider a very large system of springs and masses Mathematical Similarity Reducing PDEs Infinite sets of ODEs also result from the reduction of PDEs Finite-volume Spectral methods Finite difference Example: heat equation Mathematical Similarity Reducing PDEs Finite-volume formulation Mathematical Similarity Reducing PDEs An infinite set of differential equations Mathematical Similarity Reducing PDEs Can now reduce the continued fraction Taking the limit Now take the inverse Laplace transform Mathematical Similarity Reducing PDEs Reduced the PDE in a way that we have extracted the heat flux at the boundary Reduction process: PDE System of ODEs Single high-order ODE Single low-order ODE Alternative: Solve PDE Get a „global‟ solution Differentiate at the boundary Mathematical Similarity Applications of PDE reduction Only need a „local‟ solution Change of boundary conditions Blast furnace monitoring [Oldham and Spanier, 1974] Laser or cryogenic surgery Heat equation [Taler, 1996] Need only one thermocouple Physical self-similarity Current work: using physical self- similarity to reduce complex models Potential driven flows through bifurcating trees Self-similar equations sets can also result from physically self-similar systems Physical Similarity A self-similar model The bifurcating tree geometry Geometry seen in a wide variety of applications [Bejan, 2000] Potential-driven flow or transfer ex. heat, fluid, energy, etc. Conservation at bifurcation points q1 q q = q1+q2 q2 Physical Similarity A self-similar model Assumptions Conservation at nodes Transfer governed by a linear operator i.e., for each branch: Large system of DAE‟s 2n+1 -2 differential „branch‟ equations 2n-1 continuity equations Physical Similarity Reduction System of DAEs can be reduced as before Regular coupling from physical similarity allows for reduction For N=1 generation network: } Physical Similarity Reduction For N-generation network Physical Similarity Additional forms of similarity Similarity in operators can be used to further simplify Two forms of similarity: Similarity „within‟ a generation Symmetric networks: the operators within a generation are identical Asymmetric networks: the operators within a generation are not identical Similarity „between‟ generations Generation dependent operators depend on the generation in which the operator occurs and change between successive generations Generation independent operators do not change between generations Four possible combinations Symmetric with generation independent operators Asymmetric with generation independent operators Symmetric with generation dependent operators Asymmetric with generation dependent operators Physical Similarity Symmetric and generation independent Can further reduce the continued fraction For infinite networks: Physical Similarity Asymmetric and generation independent Further reduction: For infinite networks: Physical Similarity Symmetric and generation dependent Further reduction: For infinite networks: • Can further reduce for certain forms of Physical Similarity Asymmetric and generation dependent Further reduction: No general reduction Can reduce further for specific cases For infinite networks: Physical Similarity Applications Viscoelasticity Asymmetric, generation independent Fractional-order viscoelastic models Springs (k) and dampers (µ) [Heymans and Bauwens, 1994] Physical Similarity Applications Laminar flow through bifurcating trees Symmetric, generation dependent Using laminar pipe flow model for each branch: Using Similarity Summary of current work Representing complex engineering systems by simple mathematical models From a reductionist perspective: Mathematical systems with similarity PDEs Systems of ODEs High-order scaler ODE Lower-order scaler ODE Physical systems with similarity Use physical structure to reduce model Additional forms of similarity can offer further reduction Fractional-order system ID Current work: using fractional-order models to better fit nearly exponential experimental data System identification: building a dynamic mathematical model of a system using measured/observed data Holistic approach to modeling Grey- or black-box modeling Typically assumes integer-order models Fractional-order models can often do a better job describing real physical systems [Podlubny, 1999] ? Fractional-order system ID Nearly exponential transitions Focus on nearly exponential transitions A transition from one steady-state to another, usually the result of step change in input Typically assumed to be some combination of exponentials Commonly modeled as first or second order systems: Fractional-order models can often do a better job Fractional-order system ID Linear systems: a toy problem Consider the system System representation: U (s) 6 Y (s) s 6s 11s 6 3 2 System identification: Fractional-order system ID Linear systems: a fractance device Fractance device – an electrical circuit having properties which lie between resistance and capacitance or resistance and inductance. Has an impedance Z (i ) Construction: infinite bifurcating network of resistors and capacitors [Nakagawa and Sorimachi, 1994] N = 1: N = ∞: Fractional-order system ID Linear systems: a fractance device For N=1, 3, 6, and 9 generation fractance devices 9 N=1 Fractional-order model is a special case of the integer-order model Fractional-order system ID Nonlinear systems: a toy problem Consider the system System representation: U (s) Y (s) System identification: Fractional-order system ID Nonlinear systems: experimental results Shell-and-tube heat exchanger Steady state correlations are readily available Interested in a dynamic correlation for the response to a step input Complex systems: reductionist approach leads a complex model System of PDEs Turbulence, recirculation Very detailed Holistic or black-box: experimentally determine response Measured hot-side outlet temperature Step change in cold-side inlet flow rate Fractional-order system ID Nonlinear systems: experimental results Experimental results: Fractional-order system ID Summary of current work Representing complex engineering systems by simple mathematical models From a holistic perspective: Fractional-order models often better descriptors of complex systems than traditional integer-order models Integer-order models are special cases of fractional-order models Useful for modeling linear and non-linear systems Heat exchanger shown to be modeled well using a fractional-order model [Aoki, 2006] Future Work Work to proceed in two directions Reductionism Finding simple mathematical models of complex engineering systems that exhibit similarity properties Holism Using simple fractional-order models Future Work Reduction using similarity Reducing PDEs Approximate boundary condition transformations Pennes equation 2-D PDEs Future Work Reduction using similarity Reducing physical systems Grid-like transfer networks Future Work Reduction using fractional-order models Using fractional-order models Continue heat exchanger experiments to better develop dynamic correlation Fractional-order control: Fractional-order systems can be controlled more efficiently using fractional-order control techniques Using fractional-order system identification and experimental results to choose proper controller gains Similar to Ziegler-Nichols method of parameter tuning Based on offline time response characteristics Questions?